How can I be assured a seat on the plane?

I'm new to traveling by air (we usually drive). Tomorrow I return home via three different flights with American Airlines, and my third flight doesn't appear to have a seat for me, and it won't allow me to check in and print my boarding pass.
My ticket has a destination with a date and time on it, yet I cannot board the plane. This seems contradictory to me. How can I prevent this in the future and be assured a seat on the plane? (What about the money I spent for it...)
(Clarification: Ideally I'd prefer not upgrading to first-class or any premium/membership things.)
Best Answer
Just so you understand:
Airlines are notorious at overbooking their flights
Depending on the type of ticket you are buying you can actually guarantee a seat for yourself by reserving the seat. AA has the following information about it.
However, during the holiday season when ticket prices are generally sky high you will probably have to have a full fare ticket to be guaranteed anything.
Pictures about "How can I be assured a seat on the plane?"



Quick Answer about "How can I be assured a seat on the plane?"
How do you get the best seat every time you fly and have to choose?
Exit rows, aisle or window seats, and seats close to the front are typically considered the best seats on a plane. On a short business trip, you might want an aisle seat near the front of the plane so you can debark as quickly as possible on arrival.How can I get better seats on a plane for free?
A Simple Trick to Book a Good Airline SeatHow do I know if I will fit in an airline seat?
The Best Way of Knowing If You Will Fit in an Airline SeatAirline seats are measured in both pitch and width. Seat width is self-explanatory, but seat pitch is defined as the space between a point on one seat and the same point on the seat in front of it.6 Tips to Choose a Seat Depending on Your Flight
Sources: Stack Exchange - This article follows the attribution requirements of Stack Exchange and is licensed under CC BY-SA 3.0.
Images: Rui Chaves, Tim Gouw, Monstera, Daniel Frese